Consider the Bayesian network in Figure 1. It factorizes the joint probability into the following list of factors:
We say that this factorization is homogeneous because all the factors are combined in the same way, i.e., by multiplication.
Now suppose the 's are convergent variables. Then their conditional probabilities can be further factorized as follows:
where the factor , for instance, is the contributing factor of a to .
We say that the following list of factors
constitute a heterogeneous factorization of because the joint probability can be obtained by combining those factors in a proper order using either multiplication or the operator . The word heterogeneous is to signify the fact that different factor pairs might be combined in different ways. We call each a heterogeneous factor because it needs to be combined with the other 's by the operator before it can be combined with other factors by multiplication. In contrast, we call the factors , , and homogeneous factors.
We shall refer to that heterogeneous factorization as the heterogeneous factorization represented by the BN in Figure 1. It is obvious that this heterogeneous factorization is of finer grain than the homogeneous factorization represented by the BN.